Welcome, from another NoVA camper!

If your camper works fine plugged into the house, it should also work fine plugged into your generator.  Based on a few things you wrote, it appears you may have some misconceptions on how the power works in most modern camping trailers or your generator isn't actually outputting 120V AC.

Take a good read through here: http://www.marxrv.com/12volt/12volt.htm.  It's full of really useful information about batteries, how the electrical systems work in trailers and whatnot. 

(from the site above)

Real short version: The breaker panel, converter and fuse panel shown above are typically all part of the converter, but it's helpful to see them as separate components to show their operation.  The 120V AC power (aka "shore power") comes in through the power plug in the diagram above and into your converter.  From there, it does two things.  First, it goes out through your breaker panel to power your 120V AC accessories in your trailer.  Depending on how your trailer is outfitted, your 120V AC accessories may only be a couple of standard household style outlets or may include things such as an refrigerator (in 120V AC mode), air conditioner, microwave, TV, etc..  The other thing your converter does is charge your battery and power your 12V DC accessories.  The 12V accessories include the lights, furnace, refrigerator (in 12V DC mode, and maybe a control board depending on how "smart" it is), water heater (if it has DSI), water pump, fans, radio, LP detector, tongue jack, etc.. 

When not connected to 120V AC electrical service via the plug, your trailer is running on battery and only the 12V DC accessories will work.  You can use your generator to directly charge your battery with a special 12V charging cable, but all that does is charge your battery and won't power your 120V AC accessories.

For everything to work from your generator (both 120V AC and 12V DC), you'll need to plug your trailer's 30A "shore power" plug into your generator.  You'll likely need an adapter to make it work.  My generator has a 120V L5-30R twistlock receptacle which I think yours also has.  I had to purchase this adapter to go between the trailer's plug and my generator's receptacle.

I can power my entire trailer (120V AC - air conditioner, microwave, TV; and 12V DC -- furnace, water pump, fans, etc..) from my Yamaha ef3000ise.  Some generators can't handle the immense startup load of an air conditioner, and you may pop the breaker on the generator in that case.  No real harm, reset it and it may just be a matter of just turning off everything else while your air conditioner is running.  Otherwise, you may not be able to run your air conditioner from your generator.

There's an unlikely possibility your trailer also has an inverter (turns 12V DC battery power into 120V AC), but those are fairly rare and so far the only one I've seen in the wild is in my FIL's motorhome.  It was fairly limited in what it could power (e.g., not powerful enough for the air conditioner or microwave), and it could deplete batteries fairly quickly.

Hopefully, this got to the bottom of your question without repeating too much information you already knew.